Derivation of the Dipolar Gross--Pitaevskii Energy

Abstract

We consider $N$ trapped bosons in $\mathbb{R}^3$ interacting via a pair potential $w$ which has a long range of dipolar type. We show the convergence of the energy and of the minimizers for the many-body problem towards those of the dipolar Gross--Pitaevskii functional, when $N$ tends to infinity. In addition to the usual cubic interaction term, the latter has the long range dipolar interaction. Our results hold under the assumption that the two-particle interaction is scaled in the form $N^{3\beta - 1} w(N^{\beta} x)$ for some $0 \leq \beta < \beta_{\max}$ with $\beta_{\max} = 1/3 + s/(45+42s),$ where $s$ is related to the growth of the trapping potential.